Question: Solve for $x$: $\sqrt[3]{20x + \sqrt[3]{20x + 13}} = 13.$
Note that $f(x) = \sqrt[3]{20x + \sqrt[3]{20x + 13}}$ is an increasing function, so the solution to
\[\sqrt[3]{20x + \sqrt[3]{20x + 13}} = 13\]is unique.  Furthermore, if $\sqrt[3]{20x + 13} = 13,$ then $x$ satisfies the given equation.  Thus, $20x + 13 = 13^3 = 2197,$ so $x = \boxed{\frac{546}{5}}.$